QDifferenceEquations
QEfficientRepresentation
construct the four efficient representations of a q-hypergeometric term
Calling Sequence
Parameters
Description
Examples
References
QEfficientRepresentation[1](H, q, n)
QEfficientRepresentation[2](H, q, n)
QEfficientRepresentation[3](H, q, n)
QEfficientRepresentation[4](H, q, n)
H
-
q-hypergeometric term in q^n
q
name used as the parameter q, usually q
n
variable
Let H be a q-hypergeometric term in qn. The QEfficientRepresentation[i](H,q,n) command constructs the ith efficient representation of H of the form Hn=CαnVqnQn where C, α are constant and Qn is a product of QPochhammer-function values and their reciprocals. Additionally,
Qn has the minimal number of factors,
Vqn is a rational function which is minimal in one sense or another, depending on the particular q-rational canonical form chosen to represent the certificate of Hqn.
If i=1 then degreedenomV is minimal; if i=2 then degreenumerV is minimal; if i=3 then degreenumerV+degreedenomV is minimal, and under this condition, degreedenomV is minimal; if i=4 then degreenumerV+degreedenomV is minimal, and under this condition, degreenumerV is minimal.
If QEfficientRepresentation is called without an index, the first efficient representation is constructed.
withQDifferenceEquations:
H≔Productqk+q2qk+1qk+q5−q3qk+q4−q2q3qk+q2−1q12qk+q2−1qk+q5qk+q42q4qk+1qk+q2−1q2qk+q2−1,k=0..n−1
H≔∏k=0n−1qk+q2qk+1qk+q5−q3qk+q4−q2q3qk+q2−1q12qk+q2−1qk+q5qk+q42q4qk+1qk+q2−1q2qk+q2−1
QEfficientRepresentation1H,q,n
q66qn+q2−1q22q3+qn2q4+qn2q+qnq2+qnqn+q2−1q11qn+q2−1q10qn+q2−1q9qn+q2−1q8qn+q2−1q7qn+q2−1q6qn+q2−1q5qn+q2−1q4qn+q2−1q3qn+q2−1qq2+qn−1q2−12q6nQPochhammer1−q4+q2,q,nQPochhammer1−q5+q3,q,n2q2−12q3+12q4+12q+1q2+1q11+q2−1q10+q2−1q9+q2−1q8+q2−1q7+q2−1q6+q2−1q5+q2−1q4+q2−1q3+q2−1q2+q−1QPochhammer−q4,q,nQPochhammer−1q5,q,n
QEfficientRepresentation2H,q,n
2q5−q3+1q2+q−1q+1q2+1q4−q2+12q3−q+12q3+qnq4+qnq2−12q6nQPochhammer−q12q2−1,q,nQPochhammer−q3q2−1,q,nq5q4+1q3+qn−q2qn+q4−q22qn+1q3qn+1q2qn+1qqn+1qn+q2−1qq2+qn−1qn+q5−q3QPochhammer−1q4,q,nQPochhammer−1q5,q,n
QEfficientRepresentation3H,q,n
q2q3−q+1q4−q2+1q3+qn2q4+qn2q+qnq2+qnqn+q2−1q2q2−12q6nQPochhammer1−q5+q3,q,nQPochhammer−q12q2−1,q,nq3+12q4+12q+1q2+12q2−1q3+qn−qqn+q4−q2QPochhammer−q4,q,nQPochhammer−1q5,q,n
QEfficientRepresentation4H,q,n
2q4−q2+1q3−q+1q+1q2+1q3+qnq4+qnqn+q2−1q2q2−12q6nQPochhammer−q12q2−1,q,nQPochhammer1−q5+q3,q,nq42q2−1q4+1qn+1q3qn+1q2qn+1qqn+1q3+qn−qqn+q4−q2QPochhammer−1q5,q,nQPochhammer−1q4,q,n
RegularQPochhammerFormH,q,n
q2−12q6nQPochhammer−q12q2−1,q,nQPochhammer1−q5+q3,q,nQPochhammer−1,q,nQPochhammer−1q2,q,nQPochhammer1−q4+q2,q,nQPochhammer−q3q2−1,q,nQPochhammer−q4,q,nQPochhammer−q2q2−1,q,nQPochhammer−1q4,q,n2QPochhammer1−q2+1,q,nQPochhammer−1q5,q,n
Abramov, S.A.; Le, H.Q.; and Petkovsek, M. "Efficient Representations of (q-)Hypergeometric Terms and the Assignment Problem." Submitted.
Abramov, S.A.; Le, H.Q.; and Petkovsek, M. "Rational Canonical Forms and Efficient Representations of Hypergeometric Terms." Proc. ISSAC'2003, pp. 7-14. 2003.
See Also
QDifferenceEquations[QMultiplicativeDecomposition]
QDifferenceEquations[QObjects]
QDifferenceEquations[QRationalCanonicalForm]
QDifferenceEquations[RegularQPochhammerForm]
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